Paulo A. S. Veloso and Armando M. Haeberer A FINITARY RELATIONAL ALGEBRA FOR CLASSICAL FIRST-ORDER LOGIC. Abstract

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1 Bulletin of the Section of Logic Volume 20/2 (1991), pp reedition 2005 [original edition, pp ] Paulo A. S. Veloso and Armando M. Haeberer A FINITARY RELATIONAL ALGEBRA FOR CLASSICAL FIRST-ORDER LOGIC Abstract A simple extension of Tarski s calculus of binary relations is shown to have the expressive power of first-order logic. In contrast with Cylindric and Polyadic algebras, this extension introduces only finitely many operations and constants whose standard meaning is finitary. 1. Introduction This paper shows that a simple extension of the well-known calculus of binary relations can have the expressive power of first-order logic. This seems to have been one of Tarski s intentions in developing his Calculus of Binary Relations [5], whose expressive power, however, is weaker than that of first-order logic. Subsequent efforts to attain the expressive power of firstorder logic [4,3] led to abandoning binary relations and to introduction of infinitely many new operations. Our extended calculus was originally motivated by the needs of program derivation [2]. Programs can be regarded as solutions to problems, which involve binary input-output relations [6]. But, a calculus of programs should manipulate them as objects by means of their relational properties, rather than as sets of input-output pairs. A summary of the results presented in this paper was accepted for presentation at the 9th. International Congress of Logic, Methodology and Philosophy of Science at Uppsala, Sweden.

2 A Finitary Relational Algebra for Classical First-Order Logic 53 This paper starts by revieving Tarski s theories of binary relations and the problem of expressiveness. Then, we introduce our extension to binary relations over strings (finite sequences) and show how they can represent sets of strings. The central result shows that if a set can be defined be a first-order formula, then its representation can be denoted by a relational term. Finally, we analyse our extension, comparing it with the extant algebras of logic. 2. Tarski s theories of binary relations and the problem of expressiveness Tarski develops his Elementary Theory of Binary Relations as an extension of first-order logic by introducing variables ranging over two sorts: individuals (denoted here by x, y, z,...) and relations (denoted here by italic letters r, s, t,...). The atomic sentences are of the form r(x, y) (with intended meaning x is in relation r with y ) and r s (where the symbols denotes equality). As usual, the compound sentences are obtained from atomic ones by means of logical connectives,,,,, and the quantifiers and. In the standard models of the Elementary Theory of Binary Relations, variables on individuals range over a fixed set A, while variables on relations range over some subsets of A A. The symbols introduced by Tarski are, in our notation, (for the universal relation), 0 (for the null relation), 1 (for the indentity relation) and P (for diversity relation), as relational constants, together with the following operations on relations: (complement), (converse), + (union), (intersection), (relative addition), and ; (relative product). The symbols, 0,, + and, are called absolute or Boolean, whereas 1, P,, and ;, are called relative or Peircean. To develop his Calculus of Binary Relations Tarski [5] derives from the axioms of his Elementary Theory of Binary Relations an appropriate set of theorems whose variables are exclusively relational ones. He then takes these theorems as extralogical axioms of his Calculus of Binary Relations. These extralogical axioms can be divided into three groups: axioms of the absolute or Boolean symbols (axioms for Boolean Algebras, e.g. (r +s) t (r t) + (s t)), axioms of the relative or Peircean symbols (e.g. r; s s; r), and axioms relating absolute and relative symbols (e.g. (r; s) t 0

3 54 Paulo A. S. Veloso and Armando M. Haeberer (s; t) r 0). Shorter axiomatization have been provided, for instance, by Chin, Jónsson and Tarski. Thus the standard models of Tarski s Calculus of Binary Relations are the proper relational algebras, i.e. algebras of the form T = < R, +,,,, 1, ;, >, where < R, +,,,, 1 > is a Boolean Algebra of subsets of A A, for some set A, closed under the Peircean operations, that satisfy the axioms of his calculus. In examining the expressiveness of his Calculus of Binary Relations, Tarski [5] asks whether every property of relations, relation among relations, etc., that can be defined in his Elementary Theory of Binary Relations, can be expressed in his calculus. Tarski s answer to this question is negative. According to him, even a simple expression like: x y z u(r(x, u) r(y, u) r(z, u)) (1) cannot be expressed within his relational calculus. For instance, no sentence of the relational calculus is satisfied by exactly the same relations that satisfy expression (1). Since the Elementary Theory of Relations is an extension of first-order logic, the problem of expressiveness posed by Tarski leads to the question: can the Calculus of Binary Relations be finitely extended so as to bear the same relationship to first-order logic that Boolean Algebras bear to sentential logic? In this direction subsequent works of Chin, Henkin, Monk, Tarski and Thompson [4] led to the introduction of Cylindric Algebras. But, in doing this they abandon the algebras of binary relations in favour of algebras of sets of infinitary sequences, and likewise for Halmos s Polyadic Algebras [3]. 3. Algebras of binary relations over strings Tarski s theories of binary relations are theories of subsets of A A. We will extend them by (i) giving a structure to our universe, and (ii) introducing two new operations. To accomplish extension (i), we take our universe U as being the free monoid B over a given set B, with operation denoted by and with neuter element denoted by λ. B will be referred to as the base set of U. So, elements of U will be finite sequences (strings) [x 1,..., x n ] with

4 A Finitary Relational Algebra for Classical First-Order Logic 55 x 1,..., x n B. As for extension (ii), we introduce two new (structural) operations on relations, namely fork (denoted by ) and concatenation (denoted by ), defined by r s = {< x, y z >: < x, y > r < x, z > s} r s = {< x y, z t >: < x, z > r < y, t > s} Thus, the standard models of our Extended Calculus of Binary Relations will be algebras of the form U = < R, +,,,, 1, 1 B, 1 λ, ;,,, >, where < R, +,,,, 1 > is a proper relational algebra over U = B, 1 B is the identity relation on strings over B with length 1, and 1 λ is the identity on {λ}. It is interesting to notice that < R, +,,,, 1, ;, > is T where U plays the role of the set of points. In other words, by means of 1 B, 1 λ, ;,, and, one is able to resolve U into a hierarchy of strings (of elements of B) of increasing length. It is instructive to examine how we manage to express (1) within our formalism and how we can overcome the difficulties encountered in trying to express it in Tarski s relational calculus. The key idea is the simulation of the existential quantifier u by the relative product. If we apply this simple idea to (1), we are led to something like x y z u(r(x, u) r(u, y) r(u, z)). This is equivalent to (1), but we cannot simulate the effect of u by the relative product. The reason for this is the fact that variable u now occurs three times in the matrix of the formula. Tarski s relational calculus has no variables over individuals, and the relative product r; r consumes, so to speak, variable u. We overcome this difficulty by means of the fork (which also has a character of existential quantification, as can be seen from its definition, a viewpoint that will be used later on.). Thus, we see that the addition of extra operations and the above definition of the universe of discourse increase the expressiveness of the language of relations. Encouraged by these positive results, one is tempted to conjecture that our Extended Calculus of Binary Relations has the expressive power of first-order logic. We will show that indeed this is so.

5 56 Paulo A. S. Veloso and Armando M. Haeberer 4. Definable subsets and their representations Consider a first-order language L with equality, given by a set P of predicate symbols (we do not consider functions symbols, since functions can, for our purposes, by replaced by their graphs). As usual, a structure C for L consists of a domain C together with a realization p C C n, for each n-ary predicate symbol p in P. This can be extended to assign a realization for each formula φ Φ(L) (where Φ(L) is the set of formulae of L). It is more convenient, however, to view a formula as defining a set of strings over C. For this purpose we introduce some notation to be used also later. We let f(φ) be the set of variables with free occurrences in formula φ, and for a finite set Y of variables, we let h(y ) = max{i : v i Y }. Consider a formula φ Φ(L) with h(f(φ)) = n; we associate with φ a set φ C of strings over C defined as follows φ C = {[c 1,..., c n ] : C = φ(c 1,..., c n )} where C = φ(c 1,..., c n ) means that the assignment of c i to v i, for i = 1,..., n, satisfies φ in C [1]. We call a subset S of U = C definable iff S = φ C for some formula φ Φ(L). We shall denote by E(C) the set of all finite unions of definable subsets of U. Clearly, E(C) is a Boolean Algebra of subsets of U. A structure C for L assigns to each n-ary predicate symbol p an n- ary relation p C. In addition, we wish to regard a structure as assigning a binary relation (on strings) to each predicate symbol. Thus, we define the relational realization of predicate p in the structure C as C[p] = {< [c 1,..., c n ], λ >: < c 1,..., c n > p C }. The above relational interpretation of predicate symbols suggests using them to build other relations by means of the relational operations. The set T (L) of relational terms is obtained from the basic terms, 1, 1 C,, and p (for each p P), by means of the relational operation symbols +,,, ;,,,. (This is the set of closed terms, the set of terms with variables in W has, in addition, all variables in W as basic terms.) Now, given a structure C for L, with domain C, each such term denotes binary relation C[ ] on U = C defined inductively in the obvious way (e.g. C[ ] = C[ ].) We call a binary relation r U U denotable iff r = C[ ] for some T (L), and use D(C) to refer to the set of all denotable relations over C.

6 A Finitary Relational Algebra for Classical First-Order Logic 57 In order to correlate the elementary subsets in E(C) with the denotable relations in D(C), we need a way of representing a subset S U by a binary relation r U U. The relational interpretation of a predicate symbol suggests representing the set S U by the relation S {λ}. Let Λ(U) be the set of all binary relations r over U with Ran(r) = {λ}. We define Λ : P(U) Λ(U) by Λ(S) = S {λ}. This representation is quite natural, in that Λ(p C ) = C[p]; but another one will prove more convenient. Let I(U) be the set of all identity relations over U and define I : P(U) I(U) by assigning to S the identity relation over S. Some simple properties of these representations will be of interest. First, both representations of sets are equivalent, via a bijection b : Λ(U) I(U) and its inverse. Second, these bijections can be represent by terms, b being represented by the term b(w) = (w, ) 1. Thus, from the viewpoint of denotable relations, they are interchangeable. Finally, both representations are bijective, their inverses assigning to each such relation its domain. In fact, it is interesting to notice that x S iff < x, x > I(S). Thus, S is elementarily definable iff I(S) is so, and both by prenex formulae with the same prefix [1]. 5. Main result: expressiveness We are now ready for our main result, which will show that terms in our Extended Calculus of Binary Relations on strings have the expressive power of first-order logic. This will be established by showing that any set of strings that can be defined by a first-order formula can also be denoted be a closed term of our Extended Calculus Binary Relations on strings. In view of the remarks in the preceding section, concerning representation of subsets as relations, it suffices to show that, for every set in E(C), there exists a term in T (L), such that C[ ] is the identity on S. So, let us denote by I(L) the set of all terms in T (L), such that, for every structure C for L, C[ ] C[1]. Also, we define, by induction on m, e 1 = 1 C and e m = e m 1 1 C. Theorem Given a first-order language L, there exists a function T : Φ(L) I(L), such that for every structure C for L, I(φ C ) = C[T (φ)], for every formula φ Φ(L). Proof. We will define T by induction on the structure of formula φ.

7 58 Paulo A. S. Veloso and Armando M. Haeberer Basis: Let us distinguish three cases. Case 1: φ is v 1 v 2. Then, we set T (v 1 v 2 ) = 2; 1 C ; 2, where 2 = 1 1. Case 2: φ is p(v 1,..., v n ) with p P. Then, we set T (p(v 1,..., v n )) = (p; p) 1. Case 3: φ is an atomic formula α(u 1,..., u m ) obtained from one of the above α(v 1,..., v n ) by means of a substitution σ. Then, Lemma 2 below yields T (α(u 1,..., u m )) = s; T (α(v 1,..., v n )) = s, where s is a term, simulating σ, easily constructed by means of 1, 1 C and. Inductive step If φ is ψ, then we set T ( ψ) = T (ψ) e n. Now, let h(f(φ)) = n. In view of Lemma 1 below, it suffices to consider φ of the form φ v 1 v 1... v n v n. If φ is ψ θ, then φ is equivalent to ψ θ, where ψ is ψ v 1 v 1... v n v n and similarly for θ. Lemma 1 applied to the inductive hypothesis gives T (ψ ) and T (θ ). We then set T (ψ θ) = T (ψ ) + T (θ ). If φ is v i ψ, then clearly φ is equivalent to v n ψ v n v n, where ψ is obtained from ψ by the substitution σ that interchanges v i and v n. So, lemma 2 below, applied to the inductive hypothesis, gives T (ψ ). Then, with Π 1 = 1 C, we set T ( v n ψ ) = Π 1 ; T (ψ ); Π 1, whence Lemma 1 will give T (φ). Now, we define, by induction on n, d 0 = ; 1 λ and d n = d n 1 Π n ; where the projections Π j are defined by induction: Π 1 = 1 C and Π j = ( ; 1 C Π j 1 ). Lemma 1. For every m 0 and for every formula φ with h(f(φ)) = n, if φ is the formula φ v 1 v 1... v m+n v m+n, one can have T (φ ) = T (φ) e m and T (φ) = d n ; T (φ ); d n. Lemma 2. Given a substitution σ on {1,..., n}, there exists a term s(σ), such that for any formula φ with h(f(φ)) = n, if σ(φ) is the formula obtained by applying σ to φ, then one can take T (σ(φ)) = s(σ); T (φ); s(σ). Proof. Let h(f(φ)) = n. By Lemma 1, we may assume f(φ) = {v 1,..., v n }. Let m = h(ran(σ)) be the highest index among σ(1),..., σ(n). We may assume σ onto {1,..., m}. (For otherwise, we can replace σ by τ from {v 1,..., v n,..., v p } onto {v 1,..., v m } and, by Lemma 1, φ by ψ of the form φ v 1 v 1... v n v n... v p v p, so that τ(ψ) is equivalent

8 A Finitary Relational Algebra for Classical First-Order Logic 59 to σ(φ) v 1 v 1... v m v m which is enough in view of Lemma 1.) So, assuming σ onto {v 1,..., v n }, we set s(σ) = e m ; Π σ(1) Π σ(2)... Π σ(n) Now, the set of terms I(L) and { 1 : T (L)} have the same denotations. So, from the very definitions of the operations and constants of our extended relational calculus one can see that, for each term I(L), one has a formula φ Φ(L), with two free variables, defining it, in the sense that for every structure C for L, C[ ] = φ C. Here, C is the structure obtained from C by replacing its domain C by the corresponding free monoid C, so that now strings are assigned to variables. Thus, we have that the representations of the definable subsets of C are exactly the denotable terms in I(L). 6. An analysis of our extension We will now compare Tarski s Calculus of Binary Relations, Cylindric Algebras [4] and Polyadic Algebras [3] with our algebras U = < R, +,,,, 1, 1 B, 1 λ, ;,, >. To begin our comparative analysis, we should recall that the standard models of Tarski s Calculus of Binary Relations are algebras of the form T = < R, +,,,, 1, ;, >, whose reduct < R, +,,,, 1 > is a Boolean Algebra, R is a subset of the powerset of A A, and the result of adding ; and may be called a Peircean extension of this Boolean Algebra. As mentioned, Tarski s Calculus of Binary Relations is insufficient to attain the expressive power of first-order logic and according to Tarski [5], cannot be finitely extended, in a Peircean-like manner, to attain such expressive power. Thus, both Tarski et al. with the Cylindric Algebra, with Halmos, with the Polyadic Algebras, abandon the Peircean-like way of extending < R, +,,,, 1 > in favour of more direct attack to the problem. Both approaches maintain a Boolean Algebra over an universe of elements whose internal structure, if any, is immaterial. By imposing an infinite arity, they produce theories over sequences of infinite length restricted by a locallyfiniteness condition. Tarski et al. chose to represent the existential quantifier by infinitely many cylindrifications (one for each variable), and equations v w by means of a doubly infinite sequence of diagonal elements.

9 60 Paulo A. S. Veloso and Armando M. Haeberer Halmos, on the other hand, represents the existential quantifier by means of a single function but introduces infinitely many transformations on the index set. We should notice that both Cylindric and Polyadic Algebras are models laden with a syntactical artifact, in that the former, by means of each axis, and the latter, by means of the index set, have names for variables over individuals. Thus, neither Cylindric nor Polyadic Algebras deserve to be classified as belonging to a calculus of relations. They have a somewhat intermediate status between such calculi and elementary theories of relations. Our approach is to maintain the Peircean-like way of extending the basic Boolean Algebra. So, in addition to ; and, we introduce 1 B, 1 λ, and, which are constants and operations on binary relations. The reason why we are able to attain the expressive power of first-order logic is that, instead of extending a Boolean Algebra over B B we are extending a Boolean Algebra over U U, where U is the free monoid B over B. Thus, we introduce a hierarchy of strings at the very level of the universe U of interpretation. In virtue of this fact, an algebra U = < R, +,,,, 1, 1 B, 1 λ, ;,,, > can be properly classified as belonging to a calculus of binary relations, in Tarski s original sense [5], in that it does not require variables ranging over individuals. As an important consequence of having structured our universes, neither quantifiers nor substitutions are primitive in our approach. They are instead finitely generate from simpler operations, which is not the case in the cylindric and polyadic approaches. Notice that the argument in the proposition in section 5 hinges on stimulating the effect of an existential quantifier, the remaining cases being dealt with by means of appropriate substitutions. In this sense, the argument is close to the spirit of Polyadic Algebras. But this was merely a proof strategy to lay bare the simplicity of the ideas underlying the argument. Actually, we can denote the effect of any first-order quantification and substitution by means of relational term. Our main result yields several important consequences. For instance, a formula φ Φ(L) is valid iff T (φ) 0; and a sentence θ is valid iff T (θ) 1 λ. So, a version of our calculus can be axiomatized by finitely many equations. By translating an axiomatization of first-order logic and the equations required to prove the equalities in the proposition of section 5, one can axiomatize the I(L)-part of our calculus. In order to axiomatize the full calculus, it is easier to replace the monoid concatenation by pair formation (i.e., replace the free monoid by a free grupoid), with the obvious

10 A Finitary Relational Algebra for Classical First-Order Logic 61 adaptation of fork and concatenation [2]. For then, the replacement of relation r by (( 1 r); (1 r)) 1 enables us to represent each relation within I(L). 7. Conclusions Program derivation needs have suggested us extending Tarski s Calculus of Binary Relations by structuring its universe and adding two new operations and two new constants, which are first-order definable. In our extended calculus, substitutions and quantifications turn out to be analyzable into simpler building blocks; thus, our calculus differs essentially from Cylindric and Polyadic Algebras. The main result is the explicit construction of an interpretation of first-order logic into our calculus. Therefore, our extended calculus has the expressive power of first-order logic. References [1] H. D. Ebbinghaus, J. Flum and W. Thomas, Mathematical Logic, Springer Verlag, New York, [2] A. M. Haeberer and P. A. S. Veloso, Partial Relations for Program Derivation: Adequacy, Inevitability and Expressiveness, IFIP TC2 Working Conference on Constructing Programs from Specifications, Pacific Grove, [3] P. R. Halmos, Algebraic Logic, Chelsea Publishing Company, New York, [4] L. Henkin, J. D. Monk and A. Tarski, Cylindric Algebras Part 1, North Holland, Amsterdam (1971). [5] A. Tarski, On the calculus of relations, Journal of Symbolic Logic vol. 6 no. 3 (1941) pp [6] P. A. S. Veloso, Outline of a general theory of general problems, Philosophia Naturalis vol. 21 no. 2-4 (1984) pp Departamento de Informática Pontifica Universidade Católica do Rio de Janeiro Rua Marquês de São Vincente Rio de Janeiro RJ - Brasil